Estimates for Modified (Euclidean) Gromov-Hausdorff Distance

The Gromov-Hausdorff distance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\textrm{GH}}(X,Y)$$\end{document} is well-known to be bounded above and below by the diameters of the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y$$\end{document}. In this paper, we study the modified Gromov-Hausdorff distance and the orbits of the action of the isometry group's subgroup in Euclidean spaces. It turns out that there are similar restrictions to it, but by the Chebyshev radii of the representatives of the orbits. As a consequence, we give an estimate for the distance between the Chebyshev centers of compact sets for their optimal alignment.